IEEE SMC 1 Moving Swarm Formations through Obstacle Fields Suranga Hettiarachchi and William M. Spears Abstract— In prior work we established how artificial physics can be used to self-organize swarms of mobile robots into hexagonal formations that move toward a goal. In this paper we extend the framework to moving formations through obstacle fields. We provide important metrics of performance that allow us to (a) compare the utility of different generalized force laws in artificial physics, (b) examine trade-offs between different metrics, and (c) provide a detailed method of comparison for future researchers in this area. Index Terms— artificial physics, formations, obstacle fields I. I NTRODUCTION T HE focus of our research is to build aggregate sensor sys- tems, specifically, to design rapidly deployable, scalable, adaptive, cost-effective, and robust swarms of autonomous distributed mobile sensing robots. Our objective is to provide a scientific, yet practical, approach to the design and analysis of swarm robotic systems. Target applications for robot swarms include tracing biological/chemical hazards to their source [1]. For such applications each robot forms a grid point for performing computational fluid dynamics (CFD) calculations to follow a chemical/biological plume. Hexagonal grids have been proven to be superior to traditional rectangular grids for numerical solutions of partial differential equations, as needed for CFD. In particular, they are more efficient and effective at handling boundary conditions [2]. It is assumed that robots can sense and affect nearby robots; thus, a key challenge of this project has been how to design “local” control rules. Not only do we want the desired global swarm behavior to emerge from the local interaction between robots (i.e., self-organization), but we also would like there to be some measure of fault-tolerance i.e., the global behavior degrades very gradually if individual robots are damaged. Self-repair is also desirable, in the event of damage. Self- organization, fault-tolerance, and self-repair are precisely those principles exhibited by natural physical systems. Thus, many answers to the problems of distributed control can be found by studying the natural laws of physics. In prior work we have shown how our artificial physics framework can be used to self-organize swarms of mobile robots into hexagonal lattices that move toward a goal [3], [4], [5]. We now extend the framework to include motion through an obstacle field. Our objective is two-fold. Prior research in this area has generally focused either on a small number of robots moving through a large number of obstacles, or a large number of robots moving through a small number of obstacles [6], [7]. However, the more difficult task of moving a large number of robots in formation through a large number S. Hettiarachchi (presenting, suranga@uwyo.edu) and Dr. W. Spears (wspears@cs.uwyo.edu, 307-766-5429/766-4036 FAX) are with the CS Dept, University of Wyoming, 1000 East University Ave., Laramie, WY, 82071. of obstacles is generally not addressed. Also, proposed metrics of performance are not complete, ignoring criteria such as the number of collisions between robots and obstacles, the distribution in time of the number of robots that reach the goal, and the connectivity of the formation as it moves. Hence, one objective is to provide a more complete set of metrics from which meaningful comparisons can be made. Second, we will use these metrics, coupled with a more complete experimental methodology, to examine (a) different strategies for performing the task, and (b) trade-offs between different criteria. First, we summarize the general artificial physics frame- work. Second, we summarize our technique for optimizing force law parameters using an evolutionary algorithm (EA). Third, we explain our experimental methodology and perfor- mance criteria. Fourth, we compare two different classes of force laws using the performance criteria. We conclude with a discussion of related work and our plans for future work. II. THE ARTIFICIAL PHYSICS FRAMEWORK In our artificial physics (AP) framework, virtual physics forces drive a swarm robotics system to a desired configuration or state. The desired configuration is one that minimizes overall system potential energy, and the system acts as a molecular dynamics ( ) simulation. Each robot has position and velocity . We use a discrete- time approximation to the continuous behavior of the robots, with time-step . At each time step, the position of each robot undergoes a perturbation . The perturbation depends on the current velocity, i.e., . The velocity of each robot at each time step also changes by . The change in velocity is controlled by the force on the robot, i.e., , where is the mass of that robot and is the force on that robot. and denote the magnitude of vectors and .A frictional force is included, for self-stabilization. From the start, we intended to have our framework map easily to physical hardware, and our model reflects this design philosophy. Having a mass associated with each robot allows our simulated robots to have momentum. The frictional force allows us to model actual friction, whether it is unavoidable or deliberate, in the real robotic system. With full friction, the robots come to a complete stop between sensor readings and with no friction the robots continue to move as they sense. The time step reflects the amount of time the robots need to perform their sensor readings. If is small, the robots get readings very often, whereas if the time step is large, readings are obtained infrequently. We have also included a parameter , which provides a necessary restriction on the acceleration a robot can achieve. Also, a parameter restricts the maximum velocity of the robots (and can always be scaled appropriately with to ensure smooth path trajectories).